Find Zeros, Vertex, & Graph Quadratic Functions

Hey guys! Let's break down how to find the zeros (also called roots), vertex, and y-intercept of quadratic functions, and then how to sketch their graphs. We'll tackle three different quadratic functions step-by-step. This is super important for understanding quadratics, so let's dive in!

a) f(x) = x² - 9x + 14

Let's start with the first function: f(x) = x² - 9x + 14. Our mission is to find its zeros, vertex, y-intercept, and finally, sketch its graph. This might sound like a lot, but we'll break it down into manageable pieces.

Finding the Zeros (Roots)

The zeros of a quadratic function are the x-values where the function equals zero, i.e., f(x) = 0. Graphically, these are the points where the parabola intersects the x-axis. To find the zeros, we need to solve the quadratic equation:

x² - 9x + 14 = 0

There are a couple of ways to solve this: factoring or using the quadratic formula. Let's try factoring first. We need to find two numbers that multiply to 14 and add up to -9. Those numbers are -2 and -7. So, we can factor the equation as:

(x - 2)(x - 7) = 0

Now, set each factor equal to zero:

x - 2 = 0 or x - 7 = 0

Solving for x, we get:

x = 2 or x = 7

So, the zeros of the function f(x) = x² - 9x + 14 are x = 2 and x = 7. These are the points where the graph crosses the x-axis. Knowing the zeros is a huge step in sketching the graph.

Finding the Vertex

The vertex is the highest or lowest point on the parabola. It's the turning point of the graph. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex (h) is given by the formula:

h = -b / 2a

In our case, a = 1, b = -9, and c = 14. Plugging these values into the formula, we get:

h = -(-9) / (2 * 1) = 9 / 2 = 4.5

So, the x-coordinate of the vertex is 4.5. Now, to find the y-coordinate (k), we plug this value back into the original function:

k = f(4.5) = (4.5)² - 9(4.5) + 14 = 20.25 - 40.5 + 14 = -6.25

Therefore, the vertex of the parabola is (4.5, -6.25). This point tells us the bottom of our U-shaped graph, since the coefficient of the x² term is positive (a = 1).

Finding the Y-Intercept

The y-intercept is the point where the graph intersects the y-axis. This occurs when x = 0. To find the y-intercept, we simply plug x = 0 into the function:

f(0) = (0)² - 9(0) + 14 = 14

So, the y-intercept is (0, 14). This is where the parabola crosses the vertical axis.

Sketching the Graph

Now that we have the zeros (2 and 7), the vertex (4.5, -6.25), and the y-intercept (0, 14), we can sketch the graph.

1. Plot the zeros: (2, 0) and (7, 0).
2. Plot the vertex: (4.5, -6.25).
3. Plot the y-intercept: (0, 14).
4. Since a = 1 (positive), the parabola opens upwards. Draw a smooth U-shaped curve that passes through these points. The vertex will be the lowest point on the curve.

This sketch gives us a visual representation of the quadratic function. We can see its general shape, where it crosses the axes, and its turning point.

b) f(x) = 2x² - 3x + 5

Next up, we have the function f(x) = 2x² - 3x + 5. Let's follow the same steps to find its zeros, vertex, y-intercept, and sketch the graph.

Finding the Zeros (Roots)

We need to solve the equation:

2x² - 3x + 5 = 0

Factoring might be tricky here, so let's use the quadratic formula. The quadratic formula is a powerful tool that can solve any quadratic equation in the form ax² + bx + c = 0. It's given by:

x = [-b ± √(b² - 4ac)] / 2a

In this case, a = 2, b = -3, and c = 5. Plugging these values into the quadratic formula, we get:

x = [3 ± √((-3)² - 4 * 2 * 5)] / (2 * 2)

x = [3 ± √(9 - 40)] / 4

x = [3 ± √(-31)] / 4

Notice that we have a negative number under the square root. This means the roots are complex numbers. There are no real zeros for this function, meaning the graph does not intersect the x-axis. This is a key piece of information for our sketch.

Finding the Vertex

We use the same formula for the x-coordinate of the vertex:

h = -b / 2a

Here, a = 2 and b = -3, so:

h = -(-3) / (2 * 2) = 3 / 4 = 0.75

Now, let's find the y-coordinate by plugging h back into the function:

k = f(0.75) = 2(0.75)² - 3(0.75) + 5 = 1.125 - 2.25 + 5 = 3.875

So, the vertex is (0.75, 3.875). This point is important, even though we don't have any x-intercepts. It still represents the minimum point of the parabola since 'a' is positive.

Finding the Y-Intercept

To find the y-intercept, we set x = 0:

f(0) = 2(0)² - 3(0) + 5 = 5

So, the y-intercept is (0, 5).

Sketching the Graph

Now we can sketch the graph. We know:

1. There are no real zeros (it doesn't cross the x-axis).
2. The vertex is (0.75, 3.875).
3. The y-intercept is (0, 5).

Since a = 2 (positive), the parabola opens upwards. Plot the vertex and the y-intercept. Imagine a U-shaped curve that has its lowest point at the vertex and passes through the y-intercept. The entire graph will be above the x-axis because there are no real roots.

c) f(x) = 3x² - 7x - 6

Finally, let's tackle f(x) = 3x² - 7x - 6. We'll use the same process to find the zeros, vertex, y-intercept, and sketch the graph.

Finding the Zeros (Roots)

We need to solve:

3x² - 7x - 6 = 0

Let's try factoring this time. We need two numbers that multiply to (3 * -6) = -18 and add up to -7. Those numbers are -9 and 2. We can rewrite the middle term:

3x² - 9x + 2x - 6 = 0

Now, factor by grouping:

3x(x - 3) + 2(x - 3) = 0

(3x + 2)(x - 3) = 0

Set each factor to zero:

3x + 2 = 0 or x - 3 = 0

Solving for x:

x = -2/3 or x = 3

So, the zeros are x = -2/3 and x = 3. These are the x-intercepts of our graph.

Finding the Vertex

Using the vertex formula:

h = -b / 2a

With a = 3 and b = -7:

h = -(-7) / (2 * 3) = 7 / 6 ≈ 1.17

Now, find the y-coordinate:

k = f(7/6) = 3(7/6)² - 7(7/6) - 6 = 3(49/36) - 49/6 - 6 = 49/12 - 98/12 - 72/12 = -121/12 ≈ -10.08

Therefore, the vertex is approximately (1.17, -10.08).

Finding the Y-Intercept

Set x = 0:

f(0) = 3(0)² - 7(0) - 6 = -6

So, the y-intercept is (0, -6).

Sketching the Graph

We have:

1. Zeros: x = -2/3 and x = 3
2. Vertex: (1.17, -10.08)
3. Y-intercept: (0, -6)

Since a = 3 (positive), the parabola opens upwards. Plot the zeros, vertex, and y-intercept. Draw a U-shaped curve that passes through these points, with the vertex as the lowest point. The graph will cross the x-axis at the zeros and the y-axis at the y-intercept.

Key Takeaways

Finding the zeros, vertex, and y-intercept are essential steps in understanding and graphing quadratic functions. By breaking down the process, we can sketch accurate graphs that show the behavior of these functions. Remember these steps, and you'll be graphing quadratics like a pro in no time! Hope this helped, guys! Let me know if you have any more questions.